Physics books (both popular trade books as well as textbooks) are typically rife with examples of everyday objects traveling at some crazy fraction of the speed of light. Which is fine, and usually the problems are fun to work out, but it might lead to the idea that accelerating, say, a nickel to 99% of the speed of light is no big deal.
It’s a big deal.
First, you might try figuring out how much energy you’d need by using a familiar equation from first-year physics,
So you could plug in the mass of a nickel (5 grams, or 0.005 kg) and the speed (99% of 300,000,000 m/s) and wind up thinking you’d need about 220 trillion Joules — that’s a lot! It’s about 53 kilotons of TNT, or something like 4 Hiroshima bombs worth of energy concentrated into making the nickel go fast. But even that’s not close to the right answer, because (as you might suspect) things get more complicated close to the speed of light.
Relativistic Kinetic Energy
From the Special Relativistic articles I’ve written about so far, you might predict that the “relativistic index” gamma will make an appearance, as it usually does for anything near the speed of light. As a refresher, remember that
and increases sharply when the speed v approaches the speed of light c. For our example here, you can plug in v/c = 0.99 and see that γ = 7.1; also remember that for slow speeds compared to light, γ = 1. How do we include this correction? Here’s the “hand-wavy” argument you shouldn’t take too seriously:
Most people have seen the famous relationship
Which relates an object’s energy to its mass and a (very large) constant, the speed of light squared. The famous Einstein-ian insight was that even an object at rest relative to you has a quantity of energy imbued in it because of its rest mass.
What if it’s moving? Let’s bravely postulate that we’ll express the total energy of a moving object by multiplying the above by the “gamma-factor”. That’s sort of the theme for relativity anyway, so let’s just do it (it turns out to be the right thing to do…)
Since a moving object would have a total energy that’s just the sum of the energy of motion (kinetic!) and rest energy, then we can just write:
And then, finally, the kinetic energy itself would be
For our example, (γ - 1) is about 6.1, so the total kinetic energy is about 2.7 quintillion Joules! 12.5 times higher than our non-relativistic guess, or about 50 atomic bombs — this is starting to edge into hydrogen-bomb-yield territory. If you were trying to buy this amount of energy from, say, your local utility, it would cost you about $75 million at a (generous) rate of 10 cents per kilowatt-hour!
To get a sense of where all this is going, let’s say 99% of the speed of light is just too dang slow. You want 99.9%. So now γ = 22.4, and you need about 3.5 times the previous energy to do it. Put another way, it costs you more than twice as much energy to go from 99% to 99.9% as it did to go from zero to 99% in the first place! Things escalate rapidly.
A Better Derivation of Relativistic Kinetic Energy
If you know some calculus, we can do a little better job showing where that relativistic kinetic energy expression came from.
If you ask most physics/engineering students for the definition of force, they’ll typically respond F = ma (Newton’s 2nd Law). But that’s really not a definition — it turns out to be a good descriptive approximation to the way things behave if their mass doesn’t change and if they’re not traveling close to the speed of light.
A better definition of force says that it’s the rate of change of momentum:
But the problem is that, as usual, things get mucked up close to the speed of light. It remains to be justified in a later article that the “real” relativistic momentum is obtained just by multiplying by gamma (like we did above with energy), so relativistic force would be
Now, we usually say that the kinetic energy of a moving body (in the absence of some external potential) is equal to the work done on the body, which is adding up (integrating!) the force over some distance:
I’ll call the speed inside the integral “u” so there’s no confusion with the final speed “v”. But look at that thing — it’s a tricky integral because there’s a time derivative inside, and gamma also contains a dependence on speed. Let’s use a few tricks of our own! First, let’s get rid of that dx by changing to dt, and since u = dx/dt, we get
The idea behind all this manipulation is to see that this is now a textbook case of integration-by-parts. When I was in Calc 1 I had no idea how often I’d see this in future physics courses. This is crazy useful to keep in your toolbox:
In quantum mechanics, for example, it turns out that the first quantity on the right-hand side usually vanishes, so I was mystified when authors would flip around the integration variable and pop in a negative sign without explicitly mentioning what was going on! Anyway, the left-hand side perfectly matches what we’ve got above, so
Now the remaining integral is what calculus books call “elementary”, which means we can use a substitution trick to solve it: let’s say
so
which is comparatively easy to integrate. After substituting back and plugging in the limits,
And after some elite algebraic skillz, finally,
On Deck:
The next article I’m working on is a bit of fun with complex numbers — I’ve always thought it’s a bit unfair to call them “imaginary”, so we’ll make a simple construction of a real number out of only “imaginary” ones to make the point.
If you’re a student/teacher and want to see lots of worked examples that I like to include in my classes when I teach the “standard” University Physics 1 and 2 courses, feel free to browse the (growing) collection of 150+ videos at
And if something is especially cool and you’re inclined to leave a “tip” I’m not above coffee or pizza:
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